Question

Sketch several representative vectors in the vector field. $$ \mathbf{F}(x, y)=\mathbf{i}+\mathbf{j} $$

Short answer

We can represent the given vector field \(\mathbf{F}(x, y) =\mathbf{i}+\mathbf{j}\) by drawing parallel vectors of equal length, all pointing 45 degrees north-east, at every point in the x-y plane.

Step-by-step solution

identify vector field

First, identify the vector field given by \(\mathbf{F}(x, y) =\mathbf{i}+\mathbf{j}\). Here, for every point (x, y), the vector field returns the same vector (1,1).

Sketch the vector field

Then, sketch the vectors. Each vector points in the direction 45 degrees north-east from the point (x, y). The length of the vector is 1 since the magnitude of the vector (1,1) is 1. Place vectors at several representative points, for example at every integer coordinates (x, y).

Observe the vector field

Notice that since the vector field assigns the same vector \((1,1)\) to all points in the x-y plane regardless of their locations, the field looks the same at every point. The field consists of parallel vectors of equal length, all pointing 45 degrees north-east. There's no divergence or rotation in the field.